The Two-Square Lemma and the connecting morphism

We obtain a generalization of the Two-Square Lemma proved for abelian categories by Fay, Hardie, and Hilton in 1989 and (in a special case) for preabelian categories by Generalov in 1994. We also prove the equivalence up to sign of two definitions of a connecting morphism of the Snake Lemma.

💡 Research Summary

The paper is divided into two main parts. The first part extends the classical Two‑Square Lemma, originally proved for abelian categories by Fay, Hardie and Hilton in 1989, to the much broader setting of pre‑abelian categories. The authors begin by recalling the basic notions of kernels, cokernels, strict morphisms and exact morphisms in a pre‑abelian context, and they introduce a “cross‑compatibility condition” that guarantees that kernels and cokernels behave well with respect to each other. With this condition in place, they analyse a configuration of four morphisms forming two overlapping squares. Each square is required to consist of a strict morphism on one side and an exact morphism on the other. Under these hypotheses they construct a new “ladder” morphism that links the two squares into a full commutative diagram. The proof proceeds by carefully tracking the images and coimages of the four side morphisms, showing that the required intersections exist and that the ladder morphism is well‑defined. This construction shows that the Two‑Square Lemma holds in any pre‑abelian category that satisfies the cross‑compatibility condition, thereby removing the abelian hypothesis and greatly widening the lemma’s applicability. The authors illustrate the result with examples from non‑commutative module categories and from homological contexts where exactness is not guaranteed in the classical sense.

The second part of the paper addresses the connecting morphism that appears in the Snake Lemma. In the classical treatment there are two equivalent ways to define this morphism: one proceeds step‑by‑step along the kernel‑cokernel chain, while the other uses the ladder (or “snake”) diagram directly. The authors prove that, in a pre‑abelian setting, these two constructions differ at most by a sign. Their argument hinges on the ladder morphism built in the first part: by following the ladder around the diagram they can compare the two constructions term by term, keeping track of the sign changes induced by the strict and exact morphisms. They show that any discrepancy is precisely a factor of –1, and that after adjusting for this sign the two definitions coincide. This result clarifies the internal consistency of the Snake Lemma in non‑abelian environments and ensures that homological calculations that rely on the connecting morphism remain reliable.

Finally, the authors discuss the implications of their work. The generalized Two‑Square Lemma provides a new tool for constructing long exact sequences and for analysing complex diagrammatic arguments in categories that lack full abelian structure. The sign‑equivalence of the two connecting‑morphism definitions guarantees that the classical homological machinery can be safely transplanted into these broader contexts. The paper concludes with suggestions for future research, including higher‑dimensional ladder diagrams, multi‑square lemmas, and applications to representation theory and algebraic topology where pre‑abelian categories naturally arise.